Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Let \(f(x)=x^{2}\) and \(g(x)=2 x-5 .\) Match each function in Group I with the correct expression in Group II. $$I$$ $$(f g)(x)$$ $$\mathbf{II}$$ $$A.\quad4 x^{2}-20 x+25$$ $$B.\quad x^{2}-2 x+5$$ $$C.\quad 2 x^{2}-5$$ $$D.\quad \frac{x^{2}}{2 x-5}$$ $$E. \quad x^{2}+2 x-5$$ $$F. \quad 2 x^{3}-5 x^{2}$$

Step-by-Step Solution

Answer

\((fg)(x)\) matches with expression F: \(2x^3 - 5x^2\).

Step 1: Understanding the Problem

We need to find the expression for \((fg)(x)\), which is the product of functions \(f(x)\) and \(g(x)\). The function \(f(x) = x^2\) and \(g(x) = 2x - 5\). The product \((fg)(x)\) is obtained by multiplying these two functions.

Step 2: Compute the Product of Functions

Calculate \((fg)(x)\) by multiplying the functions: \(f(x) \times g(x)\). Substitute the functions: \((fg)(x) = (x^2) \times (2x - 5)\). Applying the distributive property, we get:\[(fg)(x) = x^2 imes 2x - x^2 imes 5 = 2x^3 - 5x^2\]

Step 3: Match with Given Expressions

We compare the computed expression \(2x^3 - 5x^2\) with the expressions in Group II: - \(A\.\ 4x^2 - 20x + 25\)- \(B\.\ x^2 - 2x + 5\)- \(C\.\ 2x^2 - 5\)- \(D\.\ \frac{x^2}{2x - 5}\)- \(E\.\ x^2 + 2x - 5\)- \(F\.\ 2x^3 - 5x^2\)The correct match is expression \(F\.\ 2x^3 - 5x^2\).

Key Concepts

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to whole number powers and multiplied by coefficients. They come in different types, often classified according to their degree, which is the highest power of the variable in the expression.
For example, in our case, we have two functions: \(f(x) = x^2\) and \(g(x) = 2x - 5\). Here:

• \(f(x)\) is a quadratic polynomial because its highest power is 2.
• \(g(x)\) is a linear polynomial since its highest power is 1.

Polynomial functions can be added, subtracted, and especially multiplied together to form new polynomial expressions. This multiplication is where the distributive property plays a big role.

Distributive Property

The distributive property is a fundamental principle in algebra that helps us simplify expressions by multiplying sums. In the context of function multiplication, we apply this law to expand expressions.
For our problem, we needed to multiply two functions: \(f(x) = x^2\) and \(g(x) = 2x - 5\). Using the distributive property, we distribute \(x^2\) across the terms inside the parentheses of \(g(x)\):

• First, multiply \(x^2\) by \(2x\) to get \(2x^3\).
• Then, multiply \(x^2\) by \(-5\) to get \(-5x^2\).

These terms combine to give us the product \((fg)(x) = 2x^3 - 5x^2\). Breaking it down this way makes the problem clearer and shows how each term is formed.

Expression Matching

Once you have computed the expression, the final step is to match it with a list of given options. This is a common exercise in algebra that enhances understanding and ensures calculation accuracy.
In our scenario, after finding the product \((fg)(x) = 2x^3 - 5x^2\), our task was to find which expression in Group II matched it:

• Option A: \(4x^2 - 20x + 25\)
• Option B: \(x^2 - 2x + 5\)
• Option C: \(2x^2 - 5\)
• Option D: \(\frac{x^2}{2x - 5}\)
• Option E: \(x^2 + 2x - 5\)
• Option F: \(2x^3 - 5x^2\)

By comparing the calculated expression to these options, we identify that Option F \(2x^3 - 5x^2\) is the correct match. This step ensures that our operation of function multiplication and the application of the distributive property were carried out correctly.